In this paper, f--Dirac--harmonic maps are studied. then, we monotonicity formulas for solutions of these equations are derived. Also, Liouvile theorem for f--Dirac--harmonic maps is proved.

In this paper we present two equivalent interpretations of curvature equation for higher dimensional Seiberg-Witten equations. To achieve this we use the generalized self-duality concept of a 2-form in various dimensions.

In this paper, we generalized E.C. Kim’ s estimates by taking in to account the trace of the divergencefree symmetric tensor non−zero. We have also shown that E.C. Kim’s estimates still valid in case of the trace of the divergencefree symmetric tensor vanished identically. In the equality case, we characterized eta−Killing spinor with Killing pair over the Sasakian spin manifolds.